Gauge Fields, Membranes, and Subdeterminant Vector Models

We present a class of classically marginal N-vector models in d = 4 and d = 3 whose scalar potentials can be written as subdeterminants of symmetric matrices. The d = 3 case can be thought of as a generalization of the scalar sector of the Bagger-Lambert- Gustavsson model. Using the Hubbard-Stratonovich transformation we calculate their effective potentials which exhibit intriguing large-N scaling behaviors. We comment on the possible relevance of our models to strings, membranes, and also to a class of novel spin systems that are based on ternary commutation relations.